Question

Integral of $\int x^{13/2}{(1+x^{5/2})}^{1/2}dx$ can be expressed as $\frac{2}{5}{(1+x^{5/2})}^{3/2}[\frac{2}{7}{(1+x^{5/2})}^2-a(1+x^b)+\frac{2}{3}]+c$
then a+(1/b) = ?

Answer 1

Substitute $1+x^{5/2}=t\Rightarrow \frac{5}{2}{x}^{3/2}dx=dt$
$\therefore \int x^{13/2}{(1+x^{5/2})}^{1/2}dx=\int {(1+x^{5/2})}^{1/2}.x^5.x^{3/2}dx$
$=\int t^{1/2}.{(t-1)}^2.\frac{2}{5}dt=\frac{2}{5}\int t^{1/2}(t^2-2t+1)dt$
$=\frac{2}{5}[\frac{2}{7}.t^{7/2}-\frac{4}{5}.t^{5/2}+\frac{2}{3}{t}^{3/2}]+c$
$=\frac{2}{5}{t}^{3/2}[\frac{2}{7}{t}^2-\frac{4}{5}t+\frac{2}{3}]+c$

$\therefore a=\frac{4}{5},b=\frac{5}{2}$
$a+\frac{1}{b}=1.2$